A mathematician friend of mine once tried to explain some stuff about his field. It started with homomorphisms on Riemann surfaces and then got well beyond my comprehension. Anyway, after he'd been talking for about five minutes he said:<p>"And this has the cardinality of the monster, and no-one knows why."<p>Mathematicians keep finding peculiar and deep relationships between their subfields that, for the most part, are not yet understood. As the article states, a proof isn't really complete until it's comprehensible.
Blows my mind that the proof shows something as tangible as 18 types + all the sporadics. Since I have absolutely no insight into the field, does anyone know if the 18 is special in any sense?<p>The difference to me between physics and maths is that the latter has the capacity to be fully deducible from the axioms, and in that sense, it's fascinating that somewhere buried in our system of numbers, groups, geometry etc. lies a set of characteristics dictating the existence of those 18 types.
This is an incredibly interesting area of mathematics, and it's disappointing that it's dying out. When I was an undergrad, I wanted to go into this field, but I was dissuaded by my advisor because of the direction it was going (fewer active researchers means fewer PhD positions, and ultimately, fewer academic jobs).<p>I'm glad that there's an effort to consolidate and simplify the proof, since as they say, it could end up effectively lost forever.
Computer proofs of the classification of finite simple groups are also being worked on.<p>Well, parts of it at least [1].<p>1: <a href="http://research.microsoft.com/en-us/news/features/gonthierproof-101112.aspx" rel="nofollow">http://research.microsoft.com/en-us/news/features/gonthierpr...</a>
Glad to see a shout-out for Ronald Solomon. His group theory course was probably my favorite math class. That's more about the material than him; we just worked our way through the first part of Rotman. But he sure didn't do anything to wreck it.<p>E.g., there was the time before class started that for some reason I went to board and led the team in classifying finite groups up to order 60. He just paused at the door when he saw that, smiled, and didn't start class until we were done.<p>(Note: The reason that could be done in a few minutes is that for the purposes of the exercise, prime numbers are trivial, and so are integers that are the product of two distinct primes. That didn't leave a lot of other cases to worry about.)
<p><pre><code> The second advantage is power: if you have proved
something about regular polyhedra, then what you
have proved automatically holds true for every
polyhedron, whether it's a cube, a tetrahedron, or
some polyhedron that you have never even heard about.
</code></pre>
Is this worded correctly/true? If I prove something is true for regular polyhedra, then I don't believe that that extends to all polyhedra since regular polyhedra are a subset of all polyhedra...